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Let $X =$ Spec$(R)$ be an affine scheme and $f:Y \rightarrow X$ be an affine scheme over $X$. Let us denote the category of affine schemes over $X$ as Aff$_X$.

In Lenstra's notes Galois Theory for Schemes (p.71 Lemma 5.10, see below) he shows that in order for $Y$ to be finite etale over $X$ it is necessary and sufficient that there exists an affine scheme $g:W\rightarrow X$ in Aff$_X$ which is surjective, flat, and finite and the pull-back $Y\times_{X}W \cong \coprod_{N}W$ for some finite set $N$.

Put another way, this theorem states (with a little more work) that $Y$ is finite etale over $X$ if and only if it is locally totally split in the fppf topology. That is, there exists an fppf coverage of $X$ such that $Y$ is totally split when pulled-back over this cover.

So we could define finite etale by saying: $Y\rightarrow X$ is finite etale if there exists an fppf cover of $X$ over which $Y$ is totally split. This is nice because it feels similar to the definition of a covering space from topology.

But there are (at least) two other natural choices of topology that we could put on Aff$_X$. Namely, the flat topology and the fpqc topology.

Question: Do each of these topologies give equivalent definitions of finite etale? i.e. is it true that:

$Y$ is flat locally totally split iff $Y$ is fpqc locally totally split iff $Y$ is fppf locally totally split?

Due to the relative "fineness" of these topologies we know that fppf locally totally split implies fpqc locally totally split implies flat locally totally split. I can prove that all three coincide if $R$ is a field. I can almost show fppf locally totally split is the same as fpqc locally totally split for an arbitrary ring; I get stuck at one step.

Any references or comments would be great! Thanks

http://websites.math.leidenuniv.nl/algebra/

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Let's say that a morphism of schemes $S' \to S$ is "$N$-split" if there is an isomorphism of $S$-schemes $S' \simeq \coprod_{N} S$.

Let $f : Y \to X$ be a morphism of schemes and suppose that there is a faithfully flat $W \to X$ such that $Y \times_{X} W \to W$ is $N$-split. We show that there is an faithfully flat, finitely presented $W' \to X$ such that $Y \times_{X} W' \to W'$ is $N$-split.

We first note that if $x \in X$ is a point and $Y \times_{X} \operatorname{Spec} \mathcal{O}_{X,x} \to \operatorname{Spec} \mathcal{O}_{X,x}$ is $N$-split, then by "standard limit arguments" there is an open neighborhood $U$ of $x$ such that $Y \times_{X} U \to U$ is $N$-split. (This is good because open immersions are finitely presented, whereas the map $\operatorname{Spec} \mathcal{O}_{X,x} \to X$ is in general not finitely presented. This also tells us that at any point we're free to localize on $X$ and assume that $R$ is a local ring.)

If we knew that $W \to X$ was a filtered inverse limit of faithfully flat, finitely presented $X$-schemes, then we'd be done by these same limit arguments, but this is false; see this note by Bhargav Bhatt or this Stacks Project page.

So instead we'll use this cover $W \to X$ to deduce nice properties about $f$ (and the fppf cover $W' \to X$ we get will have little to do with $W \to X$). Namely, since the morphism $f$ is finitely presented, flat, surjective after a faithfully flat base change on $X$, the same is true for $f$ itself; see e.g. 02L0, 02L2, 0495 respectively.

Now here's a trick: we pull back $f$ by the fppf covering $f$ and replace $f : Y \to X$ by, say, the first projection $p_{1} : Y \times_{X} Y \to Y$ (and replace the cover $W \to X$ by $Y \times_{X} W \to Y$). This reduces us to the case when $f : Y \to X$ has a section $s : X \to Y$ (in our case, this comes from the diagonal embedding $\Delta : Y \to Y \times_{X} Y$). After pulling back by $W \to X$ and working Zariski locally on the base, this section $s$ is an open and closed immersion (more precisely, if $S$ is a connected scheme, any section of $\coprod_{N} S \to S$ is isomorphic to the inclusion of one of the $N$ components $S \to \coprod_{N} S$); the property of being an open immersion descends under faithfully flat maps (see e.g. 02L3), so $s$ is an open immersion. This means we may write $Y = Y' \coprod X$ as $X$-schemes, and $Y' \to X$ has the property that it is $(N-1)$-split after a faithfully flat base change, so we replace $f$ by $Y' \to X$ and conclude by downward induction on $N$.

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  • $\begingroup$ Suppose that instead of having one faithfully flat morphism $W\rightarrow X$ we instead have a (possibly infinite) family $(W_{i} \rightarrow X)_{I}$ of morphisms each of which are flat and together form a faithful family. Does this still work? We can take the coproduct of all $W_{i}$, which will surely be faithful over $X$. But is it flat? I am not sure in the case that the family is infinite --- Also, thanks for taking the time to answer my question :) $\endgroup$
    – user256340
    Sep 15, 2019 at 22:24
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    $\begingroup$ Yes, the coproduct will be flat; you can check using the definition and the fact that the local rings of a coproduct are those of the individual components. $\endgroup$ Sep 16, 2019 at 8:12
  • $\begingroup$ Okay, great. But the coproduct will not be affine in the case that I is infinite, right? $\endgroup$
    – user256340
    Sep 16, 2019 at 9:58
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    $\begingroup$ Right, it will not be affine. $\endgroup$ Sep 16, 2019 at 11:06

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